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Results for “numerical differentiation”

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At least 577 records · Page 32

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NASA NTRS1975

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitation and rotation terms in the equations are of first order in the space variables, the pressure gradient terms are of second order, and the turbulent viscosity term is of third order. The presence of a turbulent viscosity insures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the initial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial inflow plays an important role and allows collapse to occur even when the rotation is large. An approximate solution of the governing partial differential equations is also given; the equations are used to study the spacial distributions of the density and velocity.

Deissler, R. G.↗

NASA NTRS1975

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitation and rotation terms in the equations are of first order in the space variables, the pressure gradient terms are of second order, and the turbulent viscosity term is of third order. The presence of a turbulent viscosity insures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the intial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial inflow plays an important role and allows collapse to occur even when the rotation is large. An approximate solution of the governing partial differential equations is also given, in order to study the spacial distributions of the density and velocity.

Deissler, R. G.↗

NASA NTRS1976

Thermospheric winds and exospheric temperatures from incoherent scatter radar measurements in four seasons

The time-dependent equations of neutral air motion are solved subject to three constraints: two of them are the usual upper and lower boundary conditions and the third is the value of the wind-induced ion drift at any given height. Using incoherent radar data, this procedure leads to a fast, direct numerical integration of the two coupled differential equations describing the horizontal wind components and yields time dependent wind profiles and meridional exospheric neutral temperature gradients. The diurnal behavior of the neutral wind system and of the exospheric temperature is presented for two solstice and two equinox days. The data used were obtained by the St. Santin and the Millstone Hill incoherent scatter radars. The derived geographic distributions of the exospheric temperatures are compared with those predicted by the OGO-6 empirical thermospheric model.

Antoniadis, D. A.↗

NASA NTRS1976

Analysis of surface tension driven flow in floating zone melting

Surface tension driven flow in a cylindrical melt suspended between two rods was investigated by numerical solution of the steady state differential equations for heat and momentum transfer. Radiation heating and electron beam heating were considered approximately. For small values of the driving force, one rotating ring was formed in the top half of the zone, and its mirror image in the bottom half. At larger driving forces, secondary cells form which probably would undergo oscillatory motion. The influence of Prandtl number, zone movement, and buoyancy on the convection was also studied. The primary resistance to mass transfer in the laminar regime was in the center of the zone rather than at the solid-liquid interfaces.

Chang, C. E.↗

NASA NTRS1976

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitational and rotational terms in the equations are of first order in the space variables, the pressure-gradient terms are of second order, and the turbulent-viscosity term is of third order. The presence of turbulent viscosity ensures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the initial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial flow is always inward and allows collapse to occur (axially) even when the rotation is large. An approximate solution of the governing partial differential equations is also given in order to study the spatial distributions of the density and velocity.

Deissler, R. G.↗

NASA NTRS1977

A computer software system for the generation of global ocean tides including self-gravitation and crustal loading effects

A computer software system is described which computes global numerical solutions of the integro-differential Laplace tidal equations, including dissipation terms and ocean loading and self-gravitation effects, for arbitrary diurnal and semidiurnal tidal constituents. The integration algorithm features a successive approximation scheme for the integro-differential system, with time stepping forward differences in the time variable and central differences in spatial variables. Solutions for M2, S2, N2, K2, K1, O1, P1 tidal constituents neglecting the effects of ocean loading and self-gravitation and a converged M2, solution including ocean loading and self-gravitation effects are presented in the form of cotidal and corange maps.

Estes, R. H.↗

NASA NTRS1977

A computer software system for the generation of global ocean tides including self-gravitation and crustal loading effects

Estes, R. H.↗

NASA NTRS1977

A theoretical study of heterojunction and graded band gap type solar cells

A computer program was designed for the analysis of variable composition solar cells and applied to several proposed solar cell structures using appropriate semiconductor materials. The program simulates solar cells made of a ternary alloy of two binary semiconductors with an arbitrary composition profile, and an abrupt or Gaussian doping profile of polarity n-on-p or p-on-n with arbitrary doping levels. Once the device structure is specified, the program numerically solves a complete set of differential equations and calculates electrostatic potential, quasi-Fermi levels, carrier concentrations and current densities, total current density and efficiency as functions of terminal voltage and position within the cell. These results are then recorded by computer in tabulated or plotted form for interpretation by the user.

Sutherland, J. E.↗

NASA NTRS1977

Turbulent plane Couette flow using probability distribution functions

A numerical scheme employing a combination of the discrete ordinate method and finite differences is developed for solving the one-dimensional form of Lundgren's (1967) model equation for turbulent plane Couette flow. The approach used requires no a priori assumption about the form of the turbulent distribution function, and the numerical solution is obtained directly from the governing differential equations. Two different types of boundary conditions (zero-gradient and Chapman-Enskog) for the distribution function are evaluated by comparing the numerical results with experimental data. It is found that: (1) the present approach gives convergent and stable results over a wide range of Reynolds numbers; (2) Lundgren's equation yields results that compare well with experimental data for mean velocity and skin friction in the case of simple Couette flow; (3) the zero-gradient boundary condition leads to a logarithmic flow profile; and (4) the Chapman-Enskog boundary condition provides very good agreement with experimental data when applied within the near-wall region.

Srinivasan, R.↗

NASA NTRS1978

ORACLS: A system for linear-quadratic-Gaussian control law design

A modern control theory design package (ORACLS) for constructing controllers and optimal filters for systems modeled by linear time-invariant differential or difference equations is described. Numerical linear-algebra procedures are used to implement the linear-quadratic-Gaussian (LQG) methodology of modern control theory. Algorithms are included for computing eigensystems of real matrices, the relative stability of a matrix, factored forms for nonnegative definite matrices, the solutions and least squares approximations to the solutions of certain linear matrix algebraic equations, the controllability properties of a linear time-invariant system, and the steady state covariance matrix of an open-loop stable system forced by white noise. Subroutines are provided for solving both the continuous and discrete optimal linear regulator problems with noise free measurements and the sampled-data optimal linear regulator problem. For measurement noise, duality theory and the optimal regulator algorithms are used to solve the continuous and discrete Kalman-Bucy filter problems. Subroutines are also included which give control laws causing the output of a system to track the output of a prescribed model.

Armstrong, E. S.↗

NASA NTRS1977

On the improvement of deconvolution with digitized data using a Runge-Kutta integration scheme

A relatively simple change in the treatment of the input function in numerical integration of high-order differential equations by Runge-Kutta methods provides substantial improvements in accuracy, particularly when the forcing function is in digitized form. The Runge-Kutta-Gill coefficients are modified to incorporate the changes; with pulse-type excitations, improvements on the order of 2 to 50 times greater accuracy are demonstrated.

Houghton, J. R.↗

NASA NTRS1977

Coplanar tail-chase aerial combat as a differential game

A reduced-order version of the one-on-one aerial combat problem is studied as a pursuit-evasion differential game. The coplanar motion takes place at given speeds and given maximum available turn rates, and is described by three state equations which are equivalent to the range, bearing, and heading of one aircraft relative to the other. The purpose of the study is to determine those relative geometries from which either aircraft can be guaranteed a win, regardless of the maneuver strategies of the other. Termination is specified by the tail-chase geometry, at which time the roles of pursuer and evader are known. The roles are found in general, together with the associated optimal turn maneuvers, by solution of the differential game of kind. For the numerical parameters chosen, neither aircraft can win from the majority of possible initial conditions if the other turns optimally in certain critical geometries.

Merz, A. W.↗

NASA NTRS1978

Numerical method for solution of systems of non-stationary spatially one-dimensional nonlinear differential equations

A computational scheme and a standard program is proposed for solving systems of nonstationary spatially one-dimensional nonlinear differential equations using Newton's method. The proposed scheme is universal in its applicability and its reduces to a minimum the work of programming. The program is written in the FORTRAN language and can be used without change on electronic computers of type YeS and BESM-6. The standard program described permits the identification of nonstationary (or stationary) solutions to systems of spatially one-dimensional nonlinear (or linear) partial differential equations. The proposed method may be used to solve a series of geophysical problems which take chemical reactions, diffusion, and heat conductivity into account, to evaluate nonstationary thermal fields in two-dimensional structures when in one of the geometrical directions it can take a small number of discrete levels, and to solve problems in nonstationary gas dynamics.

Morozov, S. K.↗

NASA NTRS1979

Buckling of rotating beams

The stability of a beam subjected to compressive centrifugal forces arising from steady rotation about an axis which does not pass through the clamped end of the beam is analyzed to determine the critical rotational speeds for buckling in the inplane and out-of-plane directions. The differential equations of motion are solved numerically using an integrating matrix method in combination with an eigenanalysis to determine the eigenvalues from which stability is assessed. The results clarify several differences which have been identified in the literature relating to the proper behavior of the critical rotational speed for buckling as the radius of rotation of the clamped end of the beam is reduced.

White, W. F., Jr.↗

NASA NTRS1980

Analysis of solid propellant combustion in a closed vessel including secondary reaction

A theory for combustion of solid propellants in a closed vessel is presented allowing for residual exothermic chemical reaction in the bulk of the gas in the vessel. Particular attention is given to propellants exhibiting thick gaseous flame zones such as nitrocellulose, double-base and nitramine propellants. For these, the reaction at high pressures is assumed to involve mainly the oxidation of residual hydrocarbons by NO. It is shown that the direct dynamic coupling between the exothermicity, the molecular weight reduction and the changing pressure can influence the dp/dt-p traces obtained, in a manner not directly related to mass burning rate of the solid. Energy and species conservation equations are derived for the bulk of the vessel in differential form; the system is solved numerically. The results show the effect of extended chemical reaction upon measurable combustion characteristics such as dp/dt-p and burn rate pressure exponent, demonstrating its potential importance in interpretation of closed vessel firing data, depending on the pace of the residual gas phase reactions.

Benreuven, M.↗

NASA NTRS1982

On the dynamic collapse of a column impacting a rigid surface

Results are presented of an analytical investigation on the dynamic collapse of an elastic periodically supported column. The column has an attached mass at one end and impacts a rigid surface with prescribed velocity and angle of incidence at the other end. A first-order approximate nonlinear analysis is developed in which it is assumed that only first-order nonlinear terms need be retained. Differential equations are derived and solved numerically by explicit time integration in order to determine specific ranges of the basic nondimensional parameters for which the response characteristics are markedly different. To assess modeling sophistication relative to prediction accuracy, a comparison is made of the results from this approximate analysis with those of a finite element computer code based on a convected coordinate formulation.

Housner, J. M.↗

NASA NTRS1982

Computation techniques and computer programs to analyze Stirling cycle engines using characteristic dynamic energy equations

The basic equations that are used to describe the physical phenomena in a Stirling cycle engine are the general energy equations and equations for the conservation of mass and conversion of momentum. These equations, together with the equation of state, an analytical expression for the gas velocity, and an equation for mesh temperature are used in this computer study of Stirling cycle characteristics. The partial differential equations describing the physical phenomena that occurs in a Stirling cycle engine are of the hyperbolic type. The hyperbolic equations have real characteristic lines. By utilizing appropriate points along these curved lines the partial differential equations can be reduced to ordinary differential equations. These equations are solved numerically using a fourth-fifth order Runge-Kutta integration technique.

Larson, V. H.↗

NASA NTRS1983

Robot arm dynamic model reduction for control

General methods are described by which the mathematical complexities of explicit and exact state equations of robot arms can be reduced to a simplified and compact state equation representation without introducing significant errors into the robot arm dynamic model. The model reduction methods are based on homogeneous coordinates and on the Langrangian algorithm for robot arm dynamics, and utilize matrix, vector and numeric analysis techniques. The derivation of differential vector representation of centripetal and Coriolis forces which has not yet been established in the literature is presented.

Bejczy, A. K.↗